Here’s my take on this.
I can see why recovering a deleted Reel would be important. Your most direct route is Instagram’s own built-in recovery feature. On the app, you can navigate to your profile, then go to Settings and privacy > Your activity > Recently Deleted. This section usually holds deleted posts, stories, and reels for up to 30 days, giving you a window to restore them.
On the other hand, if it’s been more than 30 days, or if you don’t see it there, the chances of recovery diminish significantly. Once Instagram permanently purges the data, it’s generally gone for good from their end. The archive feature you mentioned is useful for hiding posts you might want later, which is a good practice to avoid accidental permanent deletion in the future.